A) Given three points, how would you determine:
1. What type of triangle you have (equilateral, isosceles, scalene)
2. If it is a right triangle.
3. The equation of the altitude from a point
4. The equation of a median from a point
5. The equation of a perpendicular bisector of a line.
B) Given 4 points, what is sufficient information to determine if the object is a:
1. Parallelogram
2. Rectangle
3. Rhombus
4. Square
Homework
Do p. 182 # 5, 6, 8, 9
p. 203 # 3 (also show that the diagonals are the same length), 6, 7, 10, 12.
Enrichment Questions
1. The Chord Perpendicular Bisector Theorem (CPBT) states that the perpendicular bisector of any chord passes through the centre of the circle. Verify this algebraically.
2. Varignon's Parallelogram Theorem (VPT) states that the quadrilateral formed by joining the midpoints of adjacent sides of any quadrilateral is a parallelogram. Verify this algebraically.
3. The Side-Splitting Theorem (SST) states that the segment formed by joining the midpoints of adjacent sides in a triangle is parallel to and half the length of the third side.
Step 1 - find midpoint
Step 2 - find equation of line using midpoint and vertex
Intersection of medians is centroid.
Perpendicular Bisector
perpendicular from midpoint
Step 1 - find midpoint
Step 2 - find slope
Step 3 - find perpendicular slope
Step 4 - find equation by using slope and midpoint
Intersection of perpendicular bisector will be circumcenter, equidistant from each vertex.
Altitude
perpendicular from side to opposite vertex
Step 1 - find slope of first side
Step 2 - find perpendicular slope
Step 3 - find equation by using slope and vertex.
Altitude can be used to find the height of a triangle. Find the intersection of altitude line equation and base line equation to get a point on the base. Distance from base to vertex is height.
[[Unit 2#|Table ]]
HW. p174 #14,15, 19
2.6 Distance from a Point to a Line
1. Determine the equation of the line which is perpendicular to the given line and which passes through P.
find the slope of the given line
take the negative reciprocal of the slope
use P and m to find the equation of the line.
2. Determine the intersection point of the lines.
3. Use P and the intersection point to calculate the distance from P to the given line.
Homework
1. Find the shortest distance from the given point to the given line.
Round to the nearest tenth if necessary.
a) (2, 2) and y = x + 1 (0.7071)
b) (3, 1) and x + y = -2 (4.243)
c) (3, -1) and 2x - y + 3 = 0 (4.472)
d) (0, -1) and 5x + 2y + 3=0 (0.1857)
2. Find the distance from A (-2, -2) to the line joining B(5, 2) and C(-1, 4) to the nearest hundredth. (6.008)
Distance Formula (Level 2)
Midpoint Formula (Level 2 , Level 3)
Equation of a Circle (Level 2)
Classifying a triangle (Level 3)
Classifying a quadrilateral (Level 3)
Distance point to a line.(Level 3)
Describe steps to find centroid or circumcentre. (Level 3)
One proof like Q9 or Q11 on p220 (Level 4)
Review
Remember:
Centroid is intersection of medians, and is the point of balance in a triangle.
Circumcentre is intersection of perpendicular bisectors, and is equidistant from each vertex.
To find a median, take midpoint to opposite vertex. Use these two points to get equation of median. Find intersections of two medians to get centroid.
To find perpendicular bisector, find midpoint, find slope, then find negative reciprocal for perpendicular slope. Use midpoint and new slope to find equation of perpendicular bisector.
Find intersection of two perpendicular bisectors to find circumcentre
The distance formula is derived from the Pythagorean Theorem. It's the square root of the sum of the rise squared plus the run squared.
Distance Formula Applet
.
Pg. 151 #7 #8
Pg. 162 #2 a, b, c #3, 5 a, c #9 #14 #19
PLEASE SEE ENRICHMENT TO EXPRESSION RADICALS IN THEIR SIMPLISTIC FORM.
2.2 The Midpoint Formula
The midpoint formula is the average of the two points.
For enrichment, what would the midpoint of three points be?
HW: p. 173 # 1, 2aceg, 4, 5, 7, 9, 12
2.3 The Equation of A Circle
The equation is for a circle at the origin. For enrichment think about how to place a circle anywhere on the Cartesian plane.155 # 1bde, 4bcd, 5, 8, 9, 10, 11, 15
2.4 Shapes on a Plane
The following file has the definitions you should be familiar with along with properties of quadrilaterals.A) Given three points, how would you determine:
1. What type of triangle you have (equilateral, isosceles, scalene)
2. If it is a right triangle.
3. The equation of the altitude from a point
4. The equation of a median from a point
5. The equation of a perpendicular bisector of a line.
B) Given 4 points, what is sufficient information to determine if the object is a:
1. Parallelogram
2. Rectangle
3. Rhombus
4. Square
Homework
Do p. 182 # 5, 6, 8, 9
p. 203 # 3 (also show that the diagonals are the same length), 6, 7, 10, 12.
Enrichment Questions
1. The Chord Perpendicular Bisector Theorem (CPBT) states that the perpendicular bisector of any chord passes through the centre of the circle. Verify this algebraically.
2. Varignon's Parallelogram Theorem (VPT) states that the quadrilateral formed by joining the midpoints of adjacent sides of any quadrilateral is a parallelogram. Verify this algebraically.
3. The Side-Splitting Theorem (SST) states that the segment formed by joining the midpoints of adjacent sides in a triangle is parallel to and half the length of the third side.
2.5 Special Lines
Step 2 - find equation of line using midpoint and vertex
Step 2 - find slope
Step 3 - find perpendicular slope
Step 4 - find equation by using slope and midpoint
Step 2 - find perpendicular slope
Step 3 - find equation by using slope and vertex.
HW. p174 #14,15, 19
2.6 Distance from a Point to a Line
1. Determine the equation of the line which is perpendicular to the given line and which passes through P.
- find the slope of the given line
- take the negative reciprocal of the slope
- use P and m to find the equation of the line.
2. Determine the intersection point of the lines.3. Use P and the intersection point to calculate the distance from P to the given line.
Homework
1. Find the shortest distance from the given point to the given line.
Round to the nearest tenth if necessary.
a) (2, 2) and y = x + 1 (0.7071)
b) (3, 1) and x + y = -2 (4.243)
c) (3, -1) and 2x - y + 3 = 0 (4.472)
d) (0, -1) and 5x + 2y + 3=0 (0.1857)
2. Find the distance from A (-2, -2) to the line joining B(5, 2) and C(-1, 4) to the nearest hundredth. (6.008)
Extra Practice for Unit Test
Also do Page 197 #22HW P206 #1
P207 #4 (c)
p209 #8
p211 #10 (odd letters)
p213 #17
p216 #20
Level 4 - See Extra Practice p217
Unit Test
Distance Formula (Level 2)Midpoint Formula (Level 2 , Level 3)
Equation of a Circle (Level 2)
Classifying a triangle (Level 3)
Classifying a quadrilateral (Level 3)
Distance point to a line.(Level 3)
Describe steps to find centroid or circumcentre. (Level 3)
One proof like Q9 or Q11 on p220 (Level 4)
Review
Remember:
Centroid is intersection of medians, and is the point of balance in a triangle.
Circumcentre is intersection of perpendicular bisectors, and is equidistant from each vertex.
To find a median, take midpoint to opposite vertex. Use these two points to get equation of median. Find intersections of two medians to get centroid.
To find perpendicular bisector, find midpoint, find slope, then find negative reciprocal for perpendicular slope. Use midpoint and new slope to find equation of perpendicular bisector.
Find intersection of two perpendicular bisectors to find circumcentre